A priori error estimates for state constrained semilinear parabolic
optimal control problems
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Abstract. We consider the finite element discretization of semilinear parabolic optimization problems subject to pointwise in time constraints on mean
values of the state variable. In contrast to many results in numerical analysis of
optimization problems subject to semilinear parabolic equations, we assume
weak second order sufficient conditions. Relying on the resulting quadratic
growth condition of the continuous problem, we derive rates of convergence as
temporal and spatial mesh sizes tend to zero.
1. Introduction
This paper is concerned with optimal control problems governed by semilinear
parabolic partial differential equations (PDEs) subject to pointwise in time constraints on mean values in space of the solution to the PDE. We derive convergence
rates for a space-time discretization of the problem based on conforming finite elements in space and a discontinuous Galerkin discretization method in time. The
control variable is an Rm -vector-valued function depending on time, acting distributed in the domain. The inequality constraint on the solution of the PDE is
imposed pointwise in time and averaged in space. Extending the result of [26] to
the case of semilinear parabolic PDEs, we consider, for a time interval I = (0, T )
and a domain Ω ⊂ R2 the following problem
Z Z
Z
1
α
Minimize J(q, u) :=
(1a)
(u(t, x) − ud (t, x))2 dxdt +
q(t)T q(t)dt,
2 I Ω
2 I
where the state u(t, x) and the control q(t) = (qi )m
i=1 are coupled by the semilinear
parabolic PDE
m
X
qi (t)gi (x)
in I × Ω,
∂t u(t, x) − ∆u(t, x) + d(t, x, u(t, x)) =
i=1
(1b)
u(t, x) = 0
on I × ∂Ω,
u(0, x) = u0
in {0} × Ω,
with a monotone and smooth nonlinearity d. Further, we consider control constraints
qmin ≤ q(t) ≤ qmax
(1c)
a.e. in I,
and, for a given weighting function ω(x), state constraints
Z
(1d)
u(t, x)ω(x)dx ≤ 0
∀t ∈ [0, T ].
Ω
The precise formulation of the problem will be given in the next section.
Date: February 26, 2016.
1
2
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
This class of problems is a simplified model motivated by applications from industrial processes like cooling/heating in steel manufacturing, or tumor therapy
in biomathematics. For an extended overview of the possible applications we refer, e.g., to [14, 21]. In many of these applications the control variable depends
on finitely many parameters with fixed spatial influence but varying in time. Further, especially in cooling processes and material optimization, bounds on the state
variable and its derivatives are prescribed to avoid material failure and to preserve
product quality.
Despite all these interesting applications, the literature on a priori error estimates
for semilinear parabolic optimal control, even without state constraints, has only
a few contributions. Error estimates were derived in [23, 31] in a setting including
bilateral control constraints; in the latter the authors also discussed several control
discretization approaches. Error estimates were obtained in [12, 13] for a problem
without control and state constraints.
The lack of results for semilinear parabolic problems in the presence of state
constraints is also explained by the sparsity of results for the corresponding linear
theory. Only recently, error estimates for the space-time discretization of the state
equation in the L∞ (I, L2 (Ω)) and L∞ (I, H01 (Ω))-norm were derived in [26] and [25],
respectively. Indeed, estimates in these norms are necessary for the consideration
of constraints pointwise in time on the mean value of the state variable and its
first derivative. We would also like to point out the new result [24] where a pointwise (quasi)-best-approximation result in L∞ (I × Ω) for the discretization of linear
parabolic PDEs has been derived.
Confining ourselves to the linear parabolic setting, error estimates in the L2 -norm
for pointwise in time and space state constraints are derived in [18], while [15] is
concerned with the variational discretization approach. For control constraints, we
refer to [27, 28].
Less sparse is the literature on state constrained semilinear elliptic problems.
We refer the reader to [8, 20, 30] and the references therein.
Recently, more attention was devoted to the study of second-order optimality
conditions for state constrained parabolic optimal control problems. A well-written
survey on the state-of-the art can be found in [11].
For the case at hand, we will rely on second-order sufficient conditions (SSCs)
that were introduced in [6]. The authors, inspired by techniques from nonlinear
optimization in finite-dimensional spaces, obtained SSCs that are very close to the
necessary ones. Their analysis was limited to the one dimensional case, i.e., Ω ⊂ R1 ,
and has been extended in [14] to domains of arbitrary dimensions considering, as
in our case, Rm -vector valued controls functions depending on time only. Due to
the nature of the problem, the resulting cone of critical directions can be recasted
also from the theory of semi-infinite optimization [2].
Seminal papers for the theory of SSCs in presence of integral state constrains
are [5, 17]. The former deals with boundary controls and handles the state constraints using Ekeland’s principle. The latter considers a nonlinearity in the boundary conditions and uses concepts of semi-group theory to cope with the limitations
on the dimension of the domain. More recently, [22] has overcome the limitation in
the dimension using concepts of maximal parabolic regularity. For other contributions to the theory of SSCs in presence of state constraints, we refer to [3, 9, 33].
A priori error estimates for state constrained semilinear parabolic optimal control problems
3
In this paper one of our main aims is to use weak SSCs for the continuous
problem as derived in [6] in order to prove discretization error estimates. For elliptic
state-constrained problems, it is known that the proof of convergence requires the
quadratic growth condition for the continuous problem, only. Once the quadratic
growth for the continuous problem is given, it does not matter if the growth is given
due to weak or strong SSCs, see [30] in the elliptic case.
In the parabolic setting, if one wants to achieve a clear separation of the spatial and temporal errors, the numerical analysis has previously been done in two
steps introducing an intermediate time-discrete problem, cf., [25, 27, 28] for convex
or [31] for non-convex problems. As a consequence of this approach, a quadratic
growth condition has to hold for the time-discrete problem in order to prove an
error estimate for the fully-discrete problem. Rather than relying on an additional
assumption for each time-discrete problem, SSCs can be transfered from the continuous to the semidiscrete level if one uses a rather strong SSC, see [31]. In contrast,
weak SSCs have not been shown to be stable with respect to time discretization;
and it is not clear at all if this is possible without further assumptions.
In this paper, in favor of the more general weak SSCs, we will derive error
estimates without the use of an intermediate auxiliary problem. Our main result
is the error estimate of Theorem 33, namely the convergence
!
12
T
T
kq̄ − q̄kh k2L2 (I,Rm ) ≤ c k log + 1 + h2 log + 1
k
k
which coincides with the orders obtained for convex problem in [26].
Our technique allows to derive an estimate for the error between the continuous
and semidiscrete solution of order
12
T
kq̄ − q̄k k2L2 (I,Rm ) ≤ ck log + 1 ,
k
only depending on the time step size k. The price we pay for not transfering the
SSC is that the analogous error estimate
T
kq̄k − q̄kh k2L2 (I,Rm ) ≤ ch2 log + 1
k
can not be shown.
The paper is organized as follows: in Section 2, we give a precise definition of the
model problem sketched in (1), introduce the operators and functionals involved
in the analysis and state first and second order optimality conditions. Section
3 is devoted to the time and space discretization of the problem. In Section 4,
we derive estimates in the L∞ (I, L2 (Ω))-norm for the discretization error between
the solution of the continuous, semidiscrete and discrete state equation, extending
techniques from [26] for linear parabolic problems to the case at hand. The core of
the paper is Section 5. Extending techniques for the elliptic case presented in [30]
to the parabolic case, we derive the rate of convergence for the optimal control
problem.
2. Assumptions and analytic setting
In this section, we discuss the precise analytic setting of the problem, introduce
the main assumptions, and fix the notation.
4
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
The spatial domain Ω ⊂ R2 is a convex, bounded domain with C 2 -boundary ∂Ω
and I = (0, T ) is a given time interval. Further, for i = 1, ..., m, we consider controls
qi ∈ L2 (I) and fixed functions gi ∈ L∞ (Ω). We assume that the desired state
satisfies ud ∈ L2 (I × Ω) and the initial data satisfies u0 ∈ H01 (Ω) ∩ H 2 (Ω) ∩ C(Ω).
The state constraint is denoted by F (u) := (u, ω), where ω ∈ L∞ (Ω) is a weighting
function.
In the following, we set V := H01 (Ω),R H := L2 (Ω); (·, ·)I denotes the standard
inner product in L2 (I, H), i.e., (·, ·)I = I (·, ·)dt with associated norm k · kI , while
(·, ·) and k · k is used for L2 (Ω). Throughout the paper, c will denote a generic constant independent of the discretization parameters, that may take different values
at each appearance.
Before discussing the problem in detail, we impose the following usual assumptions on the nonlinearity, see, e.g., [35, Chapter 5, Assumption 5.6].
Assumption 1. The nonlinearity d(t, x, u) : I × Ω × R is assumed to satisfy the
following:
(i) For all u ∈ R, the nonlinearity is measurable with respect to (t, x) ∈ I × Ω.
Further, for almost every (t, x) ∈ I ×Ω it is twice continuously differentiable
with respect to u.
(ii) For u = 0, there is c > 0 such that d(t, x, u) satisfies, together with its
derivatives up to order two, the boundedness condition
kd(·, ·, 0)kL∞ (I×Ω) + k∂u d(·, ·, 0)kL∞ (I×Ω) + k∂u2 d(·, ·, 0)kL∞ (I×Ω) ≤ c.
Further, each of these satisfy a local Lipschitz condition with respect to u,
i.e., for any M > 0 there exist a constant L(M ) > 0 such that for any
|uj | ≤ M j = 1, 2 there holds
k∂ui d(·, ·, u1 ) − ∂ui d(·, ·, u2 )kL∞ (I×Ω) ≤ L(M )|u1 − u2 |,
for every i = 0, 1, 2 and almost every (t, x) ∈ I × Ω.
(iii) For all u ∈ R and for almost every (t, x) ∈ I × Ω, there holds the monotonicity condition
∂u d(t, x, u) ≥ 0.
When no confusion arises, we shorten the notation for the semilinearity from
d(·, ·, u) to d(u).
We now focus on the well-posedness of the state equation (1b). We introduce
the Hilbert space
W (0, T ) = {u ∈ L2 (I, V ), ∂t u ∈ L2 (I, V ∗ )},
and the space of admissible controls
Qad = {q ∈ L2 (I, Rm ) | qmin ≤ q(t) ≤ qmax , a.e. in I}
with qmin < qmax ∈ Rm .
Denoting with V ∗ the dual space of V , we recall that the triplet
V ,→ H ,→ V ∗
forms a Gelfand triple. Then, for u, ϕ ∈ W (0, T ) we define a bilinear form
b(u, ϕ) = (∂t u, ϕ)I + (∇u, ∇ϕ)I + (u(0), ϕ(0))
A priori error estimates for state constrained semilinear parabolic optimal control problems
5
and the weak formulation of (1b) reads: for given q ∈ L2 (I, Rm ) and initial data
u0 ∈ V ∩ H 2 (Ω) ∩ C(Ω), find u ∈ W (0, T ) satisfying
(2)
b(u, ϕ) + (d(·, ·, u), ϕ)I = (qg, ϕ)I + (u0 , ϕ(0)), ∀ϕ ∈ W (0, T ).
It is well known that (2) admits a unique solution u ∈ W (0, T ) ∩ C(I × Ω), see,
e.g., [35, Theorem 5.5]. Further, thanks to the monotonicity assumption on d(·, ·, u),
the solution u of (2) satisfies the additional regularity
(3)
u ∈ L2 (I, V ∩ H 2 (Ω)) ∩ L∞ (I × Ω) ∩ H 1 (I, L2 (Ω)),
and the following stability estimates hold, cf. [31, Proposition 2.1], justifying the
use of the L2 inner-product in the notation of (2).
Proposition 2. Let u ∈ W (0, T ) be the solution of (2) for given data q, g, u0 and
d. Then, there holds
kukL∞ (I×Ω) ≤ c kqgkL∞ (I×Ω) + ku0 kL∞ (Ω) + kd(0)kL∞ (I×Ω) ,
(4)
kukL2 (I,V ) + kukL∞ (I,V ) + k∂t ukI ≤ c kqgkI + ku0 kV + kd(0)kI .
Remark 3. We observe that the regularity of u ∈ W (0, T ) is enough to treat the
state constraint. Indeed, there holds the embedding W (0, T ) ,→ C(I, H) and we
have F : W (0, T ) → C(I) where
Z
F (u)(t) :=
u(t, x)ω(x)dx.
Ω
On the other hand, we require more regularity for the solution of (2), because
stability estimates in the norms of L∞ (I × Ω), L∞ (I, H) will come into play to
ensure Lipschitz continuity for the control-to-state map. Further, we note that
u0 ∈ V ∩ C(Ω) is enough to ensure well-posedness of the problem. The assumption
u0 ∈ H 2 (Ω) is posed to use results from [26, 28], where this regularity is required to
fully exploit the approximation property of the discontinuous Galerkin method.
Thanks to (1c), we can regard the control variable q as an element of L∞ (I, Rm ).
Then the following definitions are justified. We introduce the control-to-state map
S : L∞ (I, Rm ) → W (0, T ) ∩ C(I × Ω),
associating to any given q the solution u(q) := S(q) of (2). We denote the concatenation of the control-to-state map and the state constraint F by
G = (F ◦ S) : L∞ (I, Rm ) → C(I).
In the subsequent analysis, we will need G to be of class C 2 . This is indeed the
case, see [6].
In order to formulate the optimal control problem in reduced form, we introduce
the set of feasible controls
Qfeas = {q ∈ Qad | G(q) ≤ 0}.
Then, (1) reads
(P)
min j(q) := J(q, S(q))
s.t. q ∈ Qfeas .
Proposition 4. Assuming the existence of a feasible point, problem (P) admits
at
least one solution (q̄, ū) ∈ L∞ (I, Rm ) × W (0, T ) ∩ C(I × Ω̄) ∩ H 1 (I, H) , where
ū = S(q̄).
6
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Proof. The existence of a feasible point will be justified in the next section after
introducing a Slater type regularity condition. The additional regularity of the
control is a consequence of the box-control constraints. Then, the assertion follows
by standard arguments see, e.g., [35, Theorem 5.7].
We remark that the problem at hand is non-convex due to the presence of the
nonlinear term in the state equation. As a consequence, it is suitable to consider
local solutions as defined below.
Definition 5. A control q̄ ∈ Qf eas is a local solution in the sense of L2 (I, Rm ) if
there exists some > 0 such that there holds
j(q̄) ≤ j(q)
for all q ∈ Qfeas with kq − q̄kL2 (I,Rm ) ≤ .
We conclude the section with well-known differentiability properties of the operators and functionals involved in the analysis, referring to [35, Chapter 5] for
details.
Lemma 6. The map S : L∞ (I, Rm ) → W (0, T ) ∩ L∞ (I × Ω) is of class C 2 from
L∞ (I, Rm ) to W (0, T ). For p ∈ L∞ (I, Rm ) and for all ϕ ∈ W (0, T ), its first
0
derivative S (q)p =: vp in direction p is the solution of
b(vp , ϕ) + (∂u d(·, ·, u(q))vp , ϕ)I = (pg, ϕ)I .
(5)
00
For p1 , p2 ∈ L∞ (I, Rm ) and for all ϕ ∈ W (0, T ), the second derivative S (q)p1 p2 =:
vp1 p2 in the directions p1 , p2 is the solution of
(6)
b(vp1 p2 , ϕ) + (∂u d(·, ·, uq )vp1 p2 , ϕ)I = −(∂uu d(·, ·, uq )vp1 vp2 , ϕ)I ,
where vp1 , vp2 are given by (5).
For S and its first derivative the following Lipschitz properties hold.
Lemma 7. For p, q1 , q2 ∈ Qad , there exists a constant c > 0 such that
(7a)
kS(q1 ) − S(q2 )kI ≤ ckq1 − q2 kL2 (I,Rm ) ,
(7b)
kS(q1 ) − S(q2 )kL∞ (I,H) ≤ ckq1 − q2 kL2 (I,Rm ) ,
(7c)
kS(q1 ) − S(q2 )kL∞ (I,V ) ≤ ckq1 − q2 kL2 (I,Rm ) ,
0
0
(7d)
kS (q1 )p − S (q2 )pkI ≤ ckq1 − q2 kL2 (I,Rm ) kpkL2 (I,Rm ) ,
(7e)
kS (q1 )p − S (q2 )pkL∞ (I,H) ≤ ckq1 − q2 kL2 (I,Rm ) kpkL2 (I,Rm ) .
0
0
Proof. The claim follows from [31, Lemma 2.3] where, for q1 g, q2 g ∈ L∞ (I × Ω), it
is shown that
kS(q1 ) − S(q2 )kI ≤ ckg(q1 − q2 )kI .
Adapting the argument to the time dependent nature of the control variable for
the case at hand, we deduce
kS(q1 ) − S(q2 )kI ≤ ckgkL∞ (Ω) kq1 − q2 kL2 (I,Rm ) .
Similarly, we deduce (7b), (7d), compare with the proof of [31, Lemma 2.3].
0
0
0
To show (7e), we consider ξ := S (q1 )p − S (q2 )p and define ũ := S (q2 )p. We
note that, for any ϕ ∈ W (0, T ), ξ fulfills
(8)
b(ξ, ϕ) + ∂u d(u(q1 ))ξ, ϕ I = − ∂u d(u(q1 ))ũ − ∂u d(u(q2 ))ũ, ϕ I .
A priori error estimates for state constrained semilinear parabolic optimal control problems
7
0
It is clear that, due to the boundedness of ∂u d(·), for S (q)p there hold the same
stability estimates as for S(q), compare with (4). Then, by means of such stability
estimate in L∞ (I, H) in combination with the Lipschitz continuity of ∂u d(·), we
obtain
kξkL∞ (I,H) ≤ ck ∂u d(u(q1 )) − ∂u d(u(q2 )) ũkI
≤ cku(q1 ) − u(q2 )kL4 (I×Ω) kũkL4 (I×Ω)
≤ cku(q1 ) − u(q2 )kL∞ (I,V ) kũkL∞ (I,V )
≤ ckq1 − q2 kI kpkI ,
where we used the embedding L∞ (I, V ) ,→ L4 (I × Ω).
Corollary 8. The functional j(q) : L∞ (I, Rm ) → R is of class C 2 in L∞ (I, Rm )
and for q, p, p1 , p2 ∈ L∞ (I, Rm ) there holds
Z X
m
0
j (q)(p) =
(αqi (t) + z0 (q)gi (x))pi (t) dt,
Ω i=1
00
Z
(vp1 vp2 + αp1 p2 − z0 (q)∂uu d(x, t, u(q))vp1 vp2 ) dtdx,
j (q)p1 p2 =
ΩI
where z0 (q) ∈ W (0, T ) is the adjoint state associated with q and j, defined, for all
ϕ ∈ W (0, T ) as the unique solution of
(9)
b(ϕ, z) + (∂u d(·, ·, u(q))z, ϕ)I = (uq − ud , ϕ)I ,
and vpi , i = 1, 2 is defined as (5).
Remark 9. As observed in [35, Section 5.7.4], when the control appears quadratically in the cost functional and linearly in the state equation, then the reduced
cost functional is of class C 2 not only in L∞ (I, Rm ) but also in L2 (I, Rm ), see
also [6, Remark 2.8]. In particular, this allows the introduction of a quadratic
growth condition without two-norm discrepancy.
2.1. Optimality conditions. In this section, we discuss the optimality conditions
for our optimal control problem. In a first step, we state standard first-order necessary conditions in KKT form. Then, we introduce second-order sufficient conditions
using the approach developed in [7] for semilinear elliptic problems. Their analysis
was extended to semilinear parabolic problems in [6] for the one-dimensional case.
Indeed, in [6] control functions are from L2 (I × Ω). As a consequence, the controlto-state map is not in general twice continuously differentiable from L2 (I × Ω) to
C(I ×Ω), however, it is always the case in the one-dimensional setting. This restriction has been circumvented in [14], considering, as in this paper, controls depending
on time only.
We rely on the following linearized Slater’s regularity condition.
Assumption 10. Given a local solution q̄ of (P), we assume the existence of
qγ ∈ Qad such that
(10)
0
G(q̄) + G (q̄)(qγ − q̄) ≤ −γ < 0,
for some γ ∈ R+ .
Based on the Slater condition, we obtain first order necessary optimality conditions in KKT form, see, e.g., [3].
8
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Theorem 11. Let q̄ ∈ Qf eas be a local solution of (P) such that Assumption 10 is
satisfied, and let ū be the associated state. Then, there exists a Lagrange multiplier
µ̄ ∈ C(I)∗ and an adjoint state z̄ ∈ L2 (I, V ) such that
(11a)
b(ū, ϕ) + (d(·, ·, ū), ϕ)I = (q̄g, ϕ)I + (u0 , ϕ(0))
∀ϕ ∈ W (0, T ),
(11b)
b(ϕ, z̄) + (ϕ, ∂u d(·, ·, ū)z̄) = (ū − ud , ϕ)I + hµ̄, F (ϕ)i
∀ϕ ∈ W (0, T ),
(11c)
α(q̄, q − q̄)L2 (I) + (z̄, (q − q̄)g)I ≥ 0
(11d)
hF (ū), µ̄i = 0, µ̄ ≥ 0, F (ū) ≤ 0
∀q ∈ Qad ,
where we used the linearity of F (·), and h·, ·i denotes the duality pairing between
C(I)∗ and C(I).
To discuss SSCs, we introduce the Hamiltonian function H : R×Ω×R×R×R → R
given by
H(q, u, z) = H(t, x, q, u, z) =
m
X
1
α
qi gi − d(u) ,
(u − ud )2 + q 2 + z
2
2
i=1
suppressing the first two arguments t, x in the exposition. Moreover, the reduced
Lagrangian function is given by
L(q, µ) = j(q) + hµ, G(u)i.
Remark 12. For better readability, at each (t, x) ∈ (I × Ω), we denote by H̄, L̄
the Hamiltonian and Lagrangian function when evaluated at (q̄, ū, z̄). We note that
∂H ∂ 2 H
m
m×m
-matrix. When referring to
∂q , ∂q 2 are, respectively, an R -vector and an R
the i-th component and the (i,j)-entry, we abbreviate ∂q Hi , ∂q2 Hi,j , respectively.
We now give the cone of critical directions associated with q̄ ∈ Qfeas , following [6].
Introducing the conditions

if q̄i = qmin ,
 ≥0
≤0
if q̄Ri = qmax ,
(12)
pi (t) =
for all i = 1, ..., m

=0
if Ω ∂q H̄i dx 6= 0,
(13)
(14)
F (vp ) =
∂F
(ū)vp ≤ 0 if F (ū) = 0,
∂u
Z
F (vp )dµ̄ = 0,
Ω
where vp is defined by (5), the cone of critical direction is given by
(15)
Cq̄ = {p ∈ L2 (I, Rm ) | p satisfies (12), (13), (14)}.
After this preparation, we postulate the following second-order sufficient condition.
Assumption 13. Let q̄ ∈ Qfeas fulfill, together with the associated state ū, the adjoint state z̄, and Lagrange multipliers µ̄, the first-order optimality conditions (11).
Then, we assume
(16)
∂ 2 L̄ 2
p >0
∂2q
∀p ∈ Cq̄ \ {0}.
A priori error estimates for state constrained semilinear parabolic optimal control problems
9
Remark 14. Comparing the second-order sufficient condition of Assumption 13
with the one of [6], we observe that the assumption
∂q2 H̄i,i ≥ ξ
∀t ∈ I \ Eiν , ∀i = 1, ..., m,
where
Z
n
o
Eiν = t ∈ I ∂q H̄i dx ≥ ν
Ω
is the set of sufficiently active control constraints and ξ, ν are positive constants, is
implicitly satisfied in our setting. Indeed, since the control appears quadratically in
the cost functional and linearly in the state equation, it trivially follows ∂q2 H̄i,i =
αI > 0, where I denotes the identity operator.
With the second-order conditions at hand, we obtain the following quadratic
growth condition.
Theorem 15. Let q̄ ∈ Qfeas satisfy the first order necessary optimality conditions (11) and let Assumption 13 hold. Then, there exist constants δ, η > 0 such
that
j(q) ≥ j(q̄) + δkq − q̄k2L2 (I,Rm )
(17)
for any q ∈ Qfeas with kq − q̄kL2 (I,Rm ) ≤ η.
Proof. The proof is by contradiction and moves along the lines of [6, Theorem 4.1].
The approach there has been extended to our setting in [14, Theorem 5]. The only
difference is the C 2 -differentiability in L2 (I, Rm ) of the reduced cost functional j,
see Remark 9, leading to the quadratic growth condition (17) without two-norm
discrepancy.
3. Discretization
We briefly describe the discretization in time and space of our problem. We use
the dG(0)cG(1) method, discontinuous in time and continuous in space Galerkin
method, referring to [34] for additional details.
The control variable is discretized implicitly by the optimality conditions through
the variational discretization approach, attributed to [19].
3.1. Time discretization. We consider a partitioning of I consisting of time intervals In = (tn−1 , tn ] for n = 1, ..., N and I0 = {0}, where the times ti are such
that 0 = t0 < t1 < ... < tN −1 < tN = T . The length of the interval In is kn and we
set k = maxn kn imposing that k < T . Further, we assume the existence of strictly
positive constants a, b, k̃ such that the following technical conditions hold:
min kn ≥ ak b ,
n>0
k̃ −1 ≤
kn
≤ k̃
kn+1
∀n > 0.
We denote with P0 (In , V ) the space of piecewise constant polynomials on In with
values in V . The semidiscrete state and trial space is
Uk = Uk (V ) = ϕk ∈ L2 (I, V ) ϕk,n = ϕk |In ∈ P0 (In , V ), n = 1, ..., N ,
with inner product (·, ·)In and norm k · kIn given by the restriction ofR the usual
inner product and norm of L2 (I, H) onto the interval In , i.e., (·, ·)In = In (·, ·)dt.
10
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Our functions are piecewise constant on each interval. Thus, we can simplify
standard notation and for functions ϕk ∈ Uk we write
ϕk,n+1 = ϕ+
k,n = lim+ ϕk (tn + t) = lim+ ϕk (tn+1 − t),
t→0
t→0
[ϕk ]n = ϕk,n+1 − ϕk,n .
For uk , ϕk ∈ Uk , the semidiscrete bilinear form is defined as
B(uk , ϕ) =
N
X
(∂t uk , ϕ)In + (∇uk , ∇ϕ)I +
n=1
N
X
([uk ]n−1 , ϕn ) + (uk,1 , ϕ1 )
n=2
= (∇uk , ∇ϕ)I +
N
X
([uk ]n−1 , ϕn ) + (uk,1 , ϕ1 ),
n=2
and the semidiscrete state equation reads: given q ∈ L2 (I, Rm ), u0 ∈ H 2 (Ω) ∩ V ,
find uk = uk (q) ∈ Uk such that
(18)
B(uk , ϕk ) + (d(·, ·, uk ), ϕk )I = (qg, ϕk )I + (u0 , ϕk,1 ), ∀ϕk ∈ Uk .
Using standard arguments, one can show that (18) admits a unique solution in
Uk , see [31, Theorem 3.1] and the references therein. Further, in the subsequent
analysis we will heavily rely on the fact that the solution is also uniformly bounded
in L∞ (I × Ω) independent of k.
Proposition 16. For the solution uk ∈ Uk of (18) the following stability estimates
hold
(19)
kuk kL∞ (I×Ω) ≤ c kqgkLp (I×Ω) + ku0 kL∞ + kd(·, ·, 0)kLp (I×Ω)
kuk kL∞ (I,V ) ≤ c kqgk2I + ku0 kV + kd(·, ·, 0)kI , lo
(20)
where p > 2.
Proof. For the boundedness of kuk kL∞ (I×Ω) we refer to [31, Theorem 3.1]. The
estimate (20) follows by [31, Theorem 3.2].
As for the continuous case, we now introduce the semidiscrete control-to-state
map
Sk : L∞ (I, Rm ) → Uk ,
associating to any given q the solution uk (q) := Sk (q) of (18). The state constraint
is
Fk := (·, w) : Uk → Uk (R),
the concatenation of the control-to-state map with Fk is denoted by
Gk = (Fk ◦ Sk ) : L∞ (I, Rm ) → Uk (R),
and the set of feasible controls by
Qk,feas = {q ∈ Qad | Gk (q) ≤ 0}.
Then, the semidiscrete version of (P) reads
(Pk )
min jk (q) := J(q, Sk (q))
s.t. q ∈ Qk,feas .
Arguing as in the continuous case, we have that Sk and Gk are of class C 2 .
A priori error estimates for state constrained semilinear parabolic optimal control problems
11
Lemma 17. The operator Sk : L∞ (I, Rm ) → Uk is of class C 2 . For uk = Sk (q)
0
and p ∈ L∞ (I, Rm ), its first derivative Sk (q)p := vk,p , in direction p, is the
solution of
(21)
B(vk,p , ϕk ) + (∂u d(·, ·, uk )vk,p , ϕk )I = (pg, ϕk )I , ∀ϕk ∈ Uk .
00
For p1 , p2 ∈ L∞ (I, Rm ), its second derivative Sk (q)p1 p2 = vk,p1 p2 , in the directions
p1 , p2 , is the solution of
(22) B(vk,p1 p2 , ϕk ) + (∂u d(·, ·, uk )vk,p1 p2 , ϕk )I = −(∂uu d(·, ·, uk )vk,p1 vk,p2 , ϕk )I ,
for all ϕk ∈ Uk .
Similarly to S, also for Sk and its first derivative there holds a Lipschitz property.
Lemma 18. For q1 , q2 , p ∈ L∞ (I, Rm ) there holds
kSk (q1 ) − Sk (q2 )kI ≤ ckq1 − q2 kL2 (I,Rm ) ,
(23a)
0
0
(23b)
kSk (q1 )p − Sk (q2 )pkI ≤ ckq1 − q2 kL2 (I,Rm ) kpkL2 (I,Rm ) .
(23c)
kSk (q1 )p − Sk (q2 )pkL∞ (I,H) ≤ ckq1 − q2 kL2 (I,Rm ) kpkL2 (I,Rm ) .
0
0
Proof. The proof is analogous to the one of Lemma 7 utilizing [31, Lemma 3.1].
3.2. Space discretization. We consider a family Th of subdivisions of Ω consisting
of closed triangles or quadrilaterals (tetrahedral or hexahedral in dimension three) T
which are affine
equivalent
to their reference elements. The union of these elements
S
Ωh = int T ∈Th T is assumed to be such that the vertices on ∂Ωh are located
on ∂Ω. We assume the family Th to be quasi-uniform and shape regular in the
sense of [4] denoting by hT the diameter of T and h := maxT ∈Th hT . Then, we
define the conforming finite element space Vh ⊂ V as the space of piecewise linear
functions with respect to Th with the canonical extension v Ω\Ω ≡ 0 for any v ∈ Vh .
h
Moreover, we assume that the sequence of spatial meshes is such that the L2 projection onto Vh is stable with respect to the H 1 -norm, for conditions ensuring
this stability see, e.g., [1]. Then, the discrete state and trial spaces are given by
Ukh = Ukh (Vh ) = ϕkh ∈ L2 (I, Vh ) ϕkh,n = ϕkh |In ∈ P0 (In , Vh ), n = 1, ..., N ,
and the discrete state equation reads: for q ∈ L∞ (I, Rm ), find ukh = ukh (q) ∈ Ukh
such that
(24)
B(ukh , ϕkh ) + (d(·, ·, ukh ), ϕkh )I = (qg, ϕkh )I + (u0 , ϕkh,1 ), ∀ϕkh ∈ Ukh .
Just as in the semidiscrete case, we have the following stability estimates, see [31,
Theorem 4.1]. We remark again that the uniform boundedness of ukh independent
of the discretization parameters k, h will play a crucial role.
Proposition 19. For the solution ukh ∈ Ukh of (24) the following stability estimates holds
(25)
kukh kL∞ (I×Ω) ≤ c kqgkLp (I×Ω) + kΠh u0 kL∞ (Ω) + kd(·, ·, 0)kLp (I×Ω)
(26)
kukh kL∞ (I,V ) ≤ c kqgk2I + kΠh u0 kV + kd(·, ·, 0)kI ,
where p > 2 and Πh : V → Vh is the L2 -projection in space.
12
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Next, we introduce the discrete control-to-state map Skh : L∞ (I, Rm ) → Ukh , the
discrete state constraint Fkh : Ukh → Ukh (R), the C 2 -functional Gkh = (Fkh ◦ Skh ),
and the set of feasible controls Qkh,feas = {q ∈ Qad | Gkh (q) ≤ 0}.
The discrete problem reads
(Pkh )
min jkh (q) := J(q, Skh (q))
s.t. q ∈ Qkh,feas .
Similar to the semidiscrete case, first and second derivatives of the discrete controlto-state map Skh are defined via (21), (22), respectively, with test functions from
Ukh . Further, for Skh and its first derivative there holds the Lipschitz property
analog to Lemma 18, compare with [31, Lemma 4.1].
We formulate now standard KKT optimality conditions for problem (Pkh ). These
conditions will be justified after the introduction of an auxiliary problem in Section
5. In particular, we will show in Lemma 30 that, for k, h small enough, the Slater
point for (1) is also a Slater point for (Pkh ).
Theorem 20. Let ūkh ∈ Qkh,feas be a local solution of (Pkh ) with ūkh ∈ Ukh the
associated state. Then, under Assumption 10, for k, h sufficiently small there exists
a Lagrange multiplier µ̄kh ∈ Ukh (R)∗ ∩ C(I)∗ and an adjoint state z̄kh ∈ Ukh such
that
B(ūkh , ϕ) + (d(·, ·, ūkh ), ϕ)I = (q̄kh g, ϕ)I + (u0 , ϕkh,1 )
B(ϕ, z̄kh ) + (ϕ, ∂u d(·, ·, ūkh )z̄kh ) = (ū − ud , ϕ)I + hµ̄kh , Fkh (ϕ)i
α(q̄kh , q − q̄kh )L2 (I) + (z̄kh , (q − q̄kh )g)I ≥ 0
∀ϕ ∈ Ukh ,
∀ϕ ∈ Ukh ,
∀q ∈ Qkh,feas ,
hFkh (ūkh ), µ̄kh i = 0, µ̄ ≥ 0,
where h·, ·i denotes the duality pairing between Ukh (R)∗ and Ukh (R). Further, the
Lagrange multiplier can be represented as an element of C(I)∗ by
Z
N
X
µkh,n
v(t)dt, ∀v ∈ C(I) ∪ Ukh (R).
hv, µ̄kh i =
kn
In
n=1
4. The state equation
In this section, we are interested in the derivation of L∞ (I, H) error estimates
for the solutions of the continuous, semidiscrete and discrete state equation. The
technique behind these estimates is based on a duality argument requiring, at any
level of discretization, the introduction of auxiliary linearized problems. This approach has been used in [26] for a linear parabolic state equation. We now intend
to extend it to the semilinear parabolic case adapting an idea of [32] for semilinear
elliptic equations.
4.1. Error estimates for the temporal discretization. In a first step, we introduce the backward uncontrolled linearized counterpart of the state equation. For
a given fixed q ∈ L∞ (I, Rm ), we consider u, uk solution of (2), (18), respectively,
and we define
(
d(u(t,x))−d(uk (t,x))
if u(t, x) 6= uk (t, x)
u(t,x)−uk (t,x)
d˜ =
0
else.
Then, we consider
˜ I = 0,
−(ϕ, ∂t w)I + (∇ϕ, ∇w)I + (ϕ, dw)
(27)
w(T ) = wT ,
A priori error estimates for state constrained semilinear parabolic optimal control problems
13
for any ϕ ∈ W (0, T ) ∩ H 1 (I, H), with wT ∈ H.
Denoting by Iˆ = (0, t̂), t̂ ∈ (0, T ), a truncated time interval, we introduce
(28)
−(ϕ, ∂t ŵ)Iˆ + (∇ϕ, ∇ŵ)Iˆ + (ϕ, d˜ŵ)Iˆ = 0,
w(t̂) = wT .
Further, the semidiscrete counterpart of (27), for any ϕk ∈ Uk , reads
(29)
˜ k )I = (ϕk,N , wT ).
B(ϕk , wk ) + (ϕk , dw
Before starting, we observe that, for any ϕk ∈ Uk , the following relations hold
(30)
˜ ϕk )I
B(u − uk , ϕk ) = −(d(u) − d(uk ), ϕk )I = −((u − uk )d,
(31)
˜ I.
B(ϕk , w − wk ) = −(ϕk , (w − wk )d)
In the following analysis, we will need negative norm estimates for the error
between the solutions of (27), (28), and (29). These estimates will be used to
derive the error at the time nodal points and inside the time intervals In . Their
derivation follows exactly as in [26, Lemma 5.1, Lemma 5.2], with minor changes
due to the presence of the linearization d˜ of the semilinear term, and therefore it
is omitted. The crucial point is the boundedness of d˜ in L∞ (I × Ω) which follows
from the Lipschitz continuity of d(·) and the regularity of u, uk ∈ L∞ (I × Ω).
For the convenience of the reader, the analog to [26, Lemma 5.1, Lemma 5.2] in
our case reads as follows.
Lemma 21. For the error between the solutions w, ŵ, and wk of (27), (28),
and (29), respectively, there holds
T 21
kw − ŵkL1 (I,H)
+ kw(0) − ŵ(0)kH −2 (Ω) ≤ ck log
kwT k,
ˆ
k
T 21
kwT k.
kw − wk kL1 (I,H) + kw(0) − wk,1 kH −2 (Ω) ≤ ck log
k
With these estimates at hand, we are ready to derive the main result of the
section.
Theorem 22. For given qg ∈ L∞ (I, H) and u0 ∈ H 2 (Ω) ∩ V , let u ∈ U and
uk ∈ Uk be the solution of (2) and (18), respectively. Then, there holds
21 T
ku − uk kL∞ (I,H) ≤ ck log + 1
kqgkL∞ (I,H) + ku0 kH 2 (Ω) + kd(0)kL∞ (I×Ω) .
k
Proof. Let ek = u − uk denote the error arising from the dG(0)-time-discretization.
In every time interval, we split the error into
(32)
kek kL∞ (In ,H) ≤ ku(·) − u(tn )kL∞ (In ,H) + ku(tn ) − uk (·)kL∞ (In ,H) ,
{z
} |
{z
}
|
(a1 )
(a2 )
and we analyze the two terms (a1 ), (a2 ) separately. Then, taking the maximum over
all n = 1, ..., N , we obtain the assertion. Without loss of generality, we consider
the last time interval IN . For an arbitrary time interval In , we consider (27) on
I = (0, tn ), (28) on Iˆ = (0, t̂) for t̂ ∈ (tn−1 , tn ], and the proof follows mutatis
mutandis, observing that 0 ≤ log(tn /k) ≤ log(T /k).
14
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
(a1 ) For a generic fixed time t̂ ∈ IN , we start the derivation considering the
interpolation error u(t̂) − u(tN ).
Consider the solutions w and ŵ to (27) and (28) on Iˆ = (0, t̂), respectively, with terminal value wT = u(t̂) − u(tN ). Integration by parts in time
of (27) and (28) gives
˜ I = 0,
−(ϕ(T ), w(T )) + (ϕ(0), w(0)) + (∂t ϕ, w)I + (∇ϕ, ∇w)I + (ϕ, dw)
−(ϕ(t̂), ŵ(t̂)) + (ϕ(0), ŵ(0)) + (∂t ϕ, ŵ) ˆ + (∇ϕ, ∇ŵ) ˆ + (ϕ, d˜ŵ) ˆ = 0,
I
I
I
1
for any ϕ ∈ W (0, T ) ∩ H (I, H).
In particular, setting ϕ = u, the state equation (2) yields
˜ I = 0,
−(u(T ), w(T )) + (u(0), w(0)) + (qg, w)I − (d(u), w)I + (u, dw)
−(u(t̂), ŵ(t̂)) + (u(0), ŵ(0)) + (qg, ŵ) ˆ − (d(u), ŵ) ˆ + (u, d˜ŵ) ˆ = 0.
I
I
I
By definition w(T ) = w(t̂) = wT , subtracting the equalities above, we get
(33)
(u(t̂) − u(T ), wT ) =(u(0), ŵ(0) − w(0)) + (qg, ŵ − w)Iˆ − (qg, w)I\Iˆ
˜
+ u, (ŵ − w)d˜ Iˆ −(u, dw)
I\Iˆ
|
{z
}|
{z
}
(b1 )
(b2 )
+ (d(u), w − ŵ)Iˆ + (d(u), w)I\Iˆ .
|
{z
} |
{z
}
(b3 )
(b4 )
We analyze the terms separately.
(b1 ) Due to the stability in L∞ (I × Ω) of the solutions of (2), (18) and the
˜ L∞ (I×Ω) ≤ c. Therefore,
Lipschitz continuity of d, we observe that kdk
˜ ˆ ≤ ckukL∞ (I,H) kŵ − wk 1 ˆ .
(u, (ŵ − w)d)
I
L (I,H)
(b2 ) Exploiting again the boundedness of d˜ in L∞ (I × Ω), and |T − t̂| ≤ k,
we have
Z T
˜
˜
−(u, dw)I\Iˆ ≤ (u, dw)dt
t̂
≤ ckkukL∞ (I,H) kwkL∞ (I,H) .
ˆ H)
(b3 ) The Lipschitz property of d(u) and the boundedness of d(0) in L∞ (I,
yield
(d(u), w − ŵ)Iˆ = (d(u) − d(0), w − ŵ)Iˆ + (d(0), w − ŵ)Iˆ
≤ kd(u) − d(0)kL∞ (I,H)
kw − ŵkL1 (I,H)
ˆ
ˆ
+ kd(0)kL∞ (I,H)
kw − ŵkL1 (I,H)
ˆ
ˆ
≤ c kukL∞ (I,H)
+
kd(0)k
kw − ŵkL1 (I,H)
.
ˆ
ˆ
ˆ
L∞ (I,H)
(b4 ) Using the same argument as for (b3 ), we conclude
(d(u), w)I\Iˆ = (d(u) − d(0), w)I\Iˆ + (d(0), w)I\Iˆ
≤ ck kukL∞ (I×Ω) + kd(0)kL∞ (I×Ω) kwkL∞ (I,H) .
A priori error estimates for state constrained semilinear parabolic optimal control problems
15
Going back to (33), but now with the value wT = u(t̂) − u(T ) we obtain
∞
−2
ku(t̂) − u(T )k2 ≤ c kw − ŵkL1 (I,H)
+
k(w
−
ŵ)(0)k
+
kkwk
ˆ
L (I,H)
H (Ω)
· kqgkL∞ (I,H) + ku0 kH 2 (Ω) + kd(0)kL∞ (I×Ω)
+ kukL∞ (I,H) + kukL∞ (I×Ω) .
Using the stability of the solution w of (27), i.e., kwkL∞ (I,H) ≤ ckwT k,
see, e.g., [26, Theorem 5.3], Proposition 2, Lemma 21 and after dividing by
kwT k = ku(t̂) − u(T )k, we conclude
(34) ku(t̂) − u(T )k ≤ ck log
T
k
+1
12 kqkL∞ (I,Rm ) kgkH + ku0 kH 2 (Ω)
+ kd(0)kL∞ (I×Ω) .
(a2 ) To obtain the error of the dG(0)-discretization inside the time interval IN ,
we set wT = u(tN ) − uk,N = u(T ) − uk,N in (27) and in (29). Then, for
any φ ∈ Uk + (L2 (I, V ) ∩ H 1 (I, H)) it holds
˜ I = B(ϕ, wk ) + (ϕ, dw
˜ k )I = (ϕN , u(T ) − uk,N ).
B(ϕ, w) + (ϕ, dw)
In particular, testing the relation above with ϕ = u − uk and making use
of (30) and (31), we have
˜ I
ku(T ) − uk,N k2 = B(u − uk , w) + (u − uk , dw)
˜ wk )I + ((u − uk )d,
˜ w)I
= B(u − uk , w − wk ) − ((u − uk )d,
˜ I + ((u − uk )d,
˜ w − wk )I
= B(u, w − wk ) + (uk , (w − wk )d)
= (qg, w − wk )I + (u0 , w(0) − wk (0)) −(d(u), w − wk )I
{z
}
|
(c1 )
˜ w − w k )I ,
˜ I + ((u − uk )d,
+ (uk , (w − wk )d)
{z
} |
{z
}
|
(c2 )
(c3 )
where in the last step we used (2).
We consider the three terms (c1 ) − (c3 ) separately.
(c1 ) Observing that L∞ (I, V ) ,→ L∞ (I, H), the stability result (4) of the
solution u of (2), the Lipschitz continuity of d(·), and the boundedness
of d(0) in L∞ (I, H) yield
−(d(u), w − wk )I ≤ kd(u) − d(0)kL∞ (I,H) + kd(0)kL∞ (I,H) kw − wk kL1 (I,H)
≤ c kukL∞ (I,H) + kd(0)kL∞ (I,H) kw − wk kL1 (I,H)
≤ c kqgkI + ku0 kV + kd(0)kL∞ (I,H) kw − wk kL1 (I,H) .
(c2 ) The boundedness of d˜ in L∞ (I × Ω) and the stability result of the
semidiscrete equation of Proposition 16 yield
˜ − wk ))I ≤ kuk kL∞ (I,H) kw − wk kL1 (I,H)
(uk , d(w
≤ c kqgkI + ku0 kV + kd(0)kI kw − wk kL1 (I,H) .
16
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
(c3 ) From the Lipschitz continuity of d(·), as well as the definition and
˜ it follows
boundedness of d,
˜ − uk ), w − wk )I = (d(u) − d(uk ), w − wk )I
(d(u
= (d(u) − d(0), w − wk )I + (d(0) − d(uk ), w − wk )I
≤ c kukL∞ (I,H) + kuk kL∞ (I,H) kw − wk kL1 (I,H)
≤ c kqgkI + ku0 kV + kd(0)kI kw − wk kL1 (I,H) ,
where in the last step, we used the stability of the solutions u, uk of (2)
and (18), from Proposition 2 and Proposition 16, respectively.
Summing up, for the error inside the time interval, we obtain
(35) ku(T ) − uk,N k2 ≤ c kw − wk kLI (I,H) + kw(0) − wk (0)kH −2 (Ω)
· kqgkL∞ (I,H) + ku0 kH 2 (Ω) + kd(0)kL∞ (I,H) .
In conclusion, combining (34) with (35) and thanks to Lemma 21, we obtain the
assertion dividing by kwT k = ku(T ) − uk,N k.
4.2. Error estimates for the spatial discretization. We develop error estimates for the spatial discretization of the problem using similar steps as in the
semidiscrete case. The linearization of d now reads
(
d(uk (t,x))−d(ukh (t,x))
if uk (t, x) 6= ukh (t, x)
uk (t,x)−ukh (t,x)
dˆ =
0
else.
We remark that, thanks to the Lipschitz continuity of d(·), the linearized term dˆ is
bounded in L∞ (I × Ω).
˜ Find wkh ∈
We introduce the discrete counterpart of (27) with dˆ instead of d.
Ukh such that
(36)
ˆ kh )I = (ϕN , wT ),
B(ϕkh , wkh ) + (ϕkh , dw
for any ϕkh ∈ Ukh , with wT ∈ H.
˜ namely, find
We also consider the auxiliary problem (29) with dˆ instead of d,
wk ∈ Uk such that
(37)
ˆ k )I = (ϕk,N , wT ),
B(ϕk , wk ) + (ϕk , dw
for any ϕk ∈ Uk .
We observe that for any ϕkh ∈ Ukh the following relations hold
(38)
(39)
ˆ ϕkh )I
B(uk − ukh , ϕkh ) = −(d(uk ) − d(ukh ), ϕkh )I = −((uk − ukh )d,
ˆ I.
B(ϕkh , wk − wkh ) = −(ϕkh , (wk − wkh )d)
As for the error in the dG(0)-semidiscretization, also here we will employ a
duality argument requiring estimates for the error between the solutions of (37)
and (36). The proof follows the lines of [26, Lemma 5.8 and Lemma 5.9] with the
ˆ For the convenience of the reader,
obvious modifications due to the presence of d.
we collect such estimates from [26] below.
A priori error estimates for state constrained semilinear parabolic optimal control problems
17
Lemma 23. For the error between the solutions wk , wkh of (37), (36), respectively,
there holds
(40)
kwk,1 − wkh,1 kH −2 (Ω) + T kwk,1 − wkh,1 k ≤ ch2 kwT k.
Theorem 24. For given qg ∈ L∞ (I, H) and u0 ∈ H 2 (Ω) ∩ V , let uk ∈ Uk and
ukh ∈ Ukh be the solutions of (18) and (24), respectively. Then, there holds
T
kuk − ukh kL∞ (I,H) ≤ ch2 log + 1 kqgkL∞ (I,H) + ku0 kH 2 (Ω) + kd(0)kL∞ (I×Ω) .
k
Proof. Since both uk , ukh are constant on each time interval In , we can equivalently show the estimate on a single time interval In and with no loss of generality
we consider the last time interval only. For an arbitrary time interval In , we consider (36) and (37) on I = (0, tn ) and, noting that 0 ≤ log(tn /k) ≤ log(T /k), the
proof follows mutatis mutandis.
Proceeding as in the proof of Theorem 22, we set wT = uk,N − ukh,N in (36)
and (37). Then, using (38) and (39), we have
ˆ k )I
kuk,N − ukh,N k2 = B(uk − ukh , wk ) + (uk − ukh , dw
ˆ k − ukh ), wkh )I + (d(u
ˆ k − ukh ), wk )I
= B(uk − ukh , wk − wkh ) − (d(u
ˆ k − wkh ))I + (d(u
ˆ k − ukh ), wk − wkh )I
= B(uk , wk − wkh ) + (ukh , d(w
= (qg, wk − wkh )I + (u0 , wk,1 − wkh,1 ) −(d(uk ), wk − wkh )I
|
{z
}
(a1 )
ˆ k − wkh ))I + (d(u
ˆ k − ukh ), wk − wkh )I ,
+ (ukh , d(w
|
{z
} |
{z
}
(a2 )
(a3 )
where in the last step we used (18). We analyze the three terms separately.
(a1 ) The Lipschitz continuity of d(·) and the boundedness of d(0) in L∞ (I, H),
give
−(d(uk ), wk − wkh )I ≤ c kd(uk ) − d(0)kL∞ (I,H) + kd(0)kL∞ (I,H)
· kwk − wkh kL1 (I,H)
≤ c kuk kL∞ (I,H) + kd(0)kL∞ (I,H) kwk − wkh kL1 (I,H) .
(a2 ) Recalling that dˆ is bounded, we have
ˆ k − wkh ))I ≤ ckukh kL∞ (I,H) kwk − wkh kL1 (I,H) .
(ukh , d(w
(a3 ) For the last term, we rely again on the Lipschitz continuity of d(·) to conclude
ˆ k − ukh ), wk − wkh )I = (d(uk ) − d(ukh ), wk − wkh )I
(d(u
≤ c kuk kL∞ (I,H) + kukh kL∞ (I,H) kwk − wkh kL1 (I,H) .
We now combine the previous inequalities and, thanks to the stability estimates (20)
and (26), we obtain
kuk,N − ukh,N k2 ≤ c kwk − wkh kL1 (I,H) + kwk,1 − wkh,1 kH −2 (Ω)
· kqgkL∞ (I,H) + ku0 kH 2 (Ω) + kd(0)kL∞ (I×Ω) .
18
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Noting that the L2 -estimate in (40) remains true on shorter intervals, it follows
with τn,k = T − tn−1
kwk − wkh kL1 (I,H) ≤
N
X
n=1
−1
kn τk,n
max
n=1,...,N
τk,n kwk,n − wkh,n k
T
≤ ch2 log + 1 kwT k.
k
and, using Lemma 23, dividing by kwT k = kuk,N − ukh,N k, we obtain the assertion.
5. Convergence analysis
In this section, we focus on the main result of this paper. We show that for any
local solution q̄ of the continuous problem satisfying KKT-conditions and SSCs,
there exists a sequence of local solutions q̄kh of (Pkh ) converging to q̄. To analyze
the errors induced by the discretization, we use the so called two-way feasibility
argument, see, e.g., [16, 29]. In this method the linearized Slater point qγ from
Assumption 10 is used to construct sequences of controls (competitors) which are
feasible for the continuous and discrete problem, respectively. If the problem is
linear, these sequences of feasible competitors can be used in the first order necessary and sufficient conditions to obtain convergence of the discrete problem. In
the semilinear case, due to the presence of the linearized term, the complementary
slackness condition cannot be used as in the linear setting. Therefore, the feasible
controls have to be used in combination with second order information; in particular, in the quadratic growth condition (17) arising from the second order sufficient
conditions. This approach has been used in the recent paper [30] for the semilinear
elliptic case in combination with a localization argument as in [10]. We now intend
to extend that approach to our semilinear parabolic optimal control problem with
state constraints.
In the following analysis, we will introduce auxiliary problems in a neighborhood
of the optimal local solution q̄. To this end, we denote with r > 0 a radius, to be
chosen conveniently later, and we define
Qr := {q ∈ Qad | kq − q̄kL2 (I,Rm ) ≤ r},
Qrfeas := {q ∈ Qr | G(q) ≤ 0}.
Then, the continuous auxiliary problem reads
(Pr )
min j(q) := J(q, S(q))
s.t. q ∈ Qrfeas .
Due to the SSCs, for r sufficiently small, the unique global solution of (Pr ) coincides
with the selected local solution q̄ of (P). The value of introducing the auxiliary
problem lies in the fact that a discretization of (Pr ), following later, will provide a
sequence of solutions converging to the selected local optimum.
Assumption 25. We assume that qγ satisfying Slater’s regularity condition (10),
is close enough to q̄ ∈ Qfeas , meaning
r
(42)
kqγ − q̄kL2 (I,Rm ) ≤ .
2
The fact that qγ is in a neighborhood of q̄ is a reasonable assumption. Indeed,
as observed in [30, Section 2], given any Slater point qγ with parameter γ one can
A priori error estimates for state constrained semilinear parabolic optimal control problems
19
construct a Slater point qγr = q̄+t(qγ − q̄) close to q̄ with a parameter γ(r) = tγ ' rγ
with t = min{1, r/2kqγ − q̄k}. Hence, one has that (10) holds with γ replaced by tγ.
Further, after showing that (Pkh ) admits local solutions and thanks to the following
Lemma 30, we will see that it is reasonable to assume that (42) holds also for the
discrete problem (Pkh ), namely
r
(43)
kqγ − q̄kh kL2 (I,Rm ) ≤ .
2
We will now define three constants c1 , c2 , c3 , independent of the discretization
parameter k, h. These constants are given by
T
T
12
sup k(ω, ukh (q) − u(q))kL∞ (I) ≤ c1 k ln + 1 + h2 ln + 1 ,
k
k
q∈B r (q̄)
2
00
kG (qk )kL(L2 (I,Rm )2 ;L∞ (I)) ,
sup
q∈B r (q̄)
sup
00
kGkh (qk )kL(L2 (I,Rm )2 ;L∞ (I)) ≤ c2 ,
q∈B r (q̄)
2
2
21
T
r2 T
k(G0kh (q) − G0 (q̄))(qγ − q̄)kL∞ (I) ≤ c3 k ln + 1 + h2 ln + 1 +
,
k
k
2
q∈B r (q̄)
sup
2
where B r2 (q̄) denotes an L2 (I, Rm ) ball centered in q̄ with radius 2r .
Remark 26. To see that these constants are independent of k, h proceed as follows
• For the constant c1 , we notice that this error can be estimated by the discretization errors obtained by Theorem 22, 24 noting that by the proof of
these theorems the constant in the error estimates remains bounded on
B r2 (q̄).
• The constant c2 is a consequence of G being a C 2 functional together with
a discretization error bound for G00kh .
• For the constant c3 , we notice that
Z
F (ϕ) = Fkh (ϕ) =
ϕ(t, x)ω(x)dx, ϕ ∈ W (0, T ) ∪ Ukh
Ω
is linear and consequently the error satisfies
0
0
(G0kh (q) − G0 (q̄))(qγ − q̄) = Fkh Skh (q)(qγ − q̄) − F S (q̄)(qγ − q̄)
0
0
= ω, Skh (q) − S (q̄) (qγ − q̄)
0
0
0
0
= ω, Skh (q) − S (q) + S (q) − S (q̄) (qγ − q̄)
0
0
≤ c k Skh (q) − S (q) (qγ − q̄)kL∞ (I,H)
+ kq − q̄kL2 (I,Rm ) kqγ − q̄kL2 (I,Rm ) ,
where in the last step we used the stability of S”, i.e., (7e). The remaining
term is a discretization error that can be estimated by [26, Corollary 5.5,
5.11]. Namely, we have
12
0
0
T
T
r
r
k Skh (q̄kh
) − S (q̄kh
) (qγ − q̄)kL∞ (I,H) ≤ c k log + 1 + h2 log + 1 ·
k
k
· kgkL∞ (Ω) kqγ − q̄kL∞ (I,Rm ) .
20
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
By virtue of the control constraints, we have kqγ − q̄kL∞ (I,Rm ) ≤ |qmax −
r
qmin |. Then, thanks to (42) and the feasibility of q̄kh
for (Prkh ), we conclude
!
12
r2
T
T
0
0
2
|(Gkh (q) − G (q̄))(qγ − q̄)| ≤ c3 k log + 1 + h log + 1 +
.
k
k
2
Moreover, by the above arguments it is clear that c1 –c3 remain bounded as r → 0.
As we have seen in the discussion after Assumption 25 it holds γ(r) ' rγ. Hence
there exists r̃ ≤ r such that
c3 2
3
r̃ ≤ − γ(r̃).
(44)
−γ(r̃) + c2 +
2
4
We can now summarize our requirements on r. Throughout the rest of the paper
we rely on the following.
Assumption 27. Let the radius r > 0 be small enough such that (44) holds and
the quadratic growth condition (17) holds for elements in Qrfeas . Namely,
j(q) ≥ j(q̄) + δkq − q̄k2L2 (I,Rm )
for any q ∈ Qrfeas .
After this preparation, for Gkh = Fkh ◦ Skh , we introduce the discrete auxiliary
problem (Prkh )
(Prkh )
min jkh (qkh ) := J(Skh (qkh , qkh ))
s.t. qkh ∈ Qrkh,feas := {qkh ∈ Qr | Gkh (qkh ) ≤ 0}.
We remark again that the control is not discretized, the index k, h is taken only to
clarify the association to the problem (Prkh ).
In a first step, we construct feasible competitors for (Prkh ).
Proposition 28. Let q̄ be a local solution of (P) and qγ be the Slater’s point from
Assumption 10. Let
t(k, h) =
c1 (k(log(T /k) + 1)1/2 + h2 (log(T /k) + 1))
c4 r 2 − γ
be given with c4 such that 0 < c4 r2 − γ < γ/2. Then, the sequence of controls
defined by
(45)
qt(k,h) = q̄ + t(k, h)(qγ − q̄)
is feasible for (Prkh ), for k, h sufficiently small such that 0 < t(k, h) < 1.
Proof. To verify the feasibility of qt(k,h) we use a Taylor’s expansion argument. The
definition of qt(k,h) suggests to expand G(qt(k,h) ) at q̄, obtaining
0
1 00
G(qt(k,h) ) = G(q̄) + G (q̄)(qt(k,h) − q̄) + G (qζ )(qt(k,h) − q̄)2 ,
2
where qζ is a convex combination of qt(k,h) and q̄.
A priori error estimates for state constrained semilinear parabolic optimal control problems
21
We insert this expansion in the following calculations
Gkh (qt(k,h) ) = Gkh (qt(k,h) ) − G(qt(k,h) ) + G(qt(k,h) )
0
= Gkh (qt(k,h) ) − G(qt(k,h) ) + G(q̄) + G (q̄)(qt(k,h) − q̄)
1 00
+ G (qζ )(qt(k,h) − q̄)2
2
= Gkh (qt(k,h) ) − G(qt(k,h) ) + G(q̄) + t(k, h)G(q̄) − t(k, h)G(q̄)
0
1 00
+ t(k, h)G (q̄)(qγ − q̄) + G (qζ )(qt(k,h) − q̄)2
2
= Gkh (qt(k,h) ) − G(qt(k,h) )
{z
}
|
(a1 )
0
+ (1 − t(k, h))G(q̄) + t(k, h)(G(q̄) + G (q̄)(qγ − q̄))
|
{z
}
(a2 )
1
+ G (qζ )(qt(k,h) − q̄)2 .
|2
{z
}
00
(a3 )
(a1 ) By definition of c1 it holds
Gkh (qt(k,h) ) − G(qt(k,h) ) = (ukh (qt(k,h) ) − u(qt(k,h) ), ω(x))I
12
T
T
≤ c1 k log + 1 + h2 log + 1 .
k
k
(a2 ) This part is handled thanks to the feasibility of q̄ for (P) and Slater’s
regularity condition of Assumption 10. Indeed, for k, h sufficiently small,
such that 0 < t(k, h) < 1, we have
(1 − t(k, h))G(q̄) ≤ 0,
0
t(k, h)(G(q̄) + G (q̄)(qγ − q̄)) ≤ −t(k, h)γ,
from which we obtain
(a2 ) ≤ −t(k, h)γ.
(a3 ) By definition of c2 it follows
00
G (qζ )(qt(k,h) − q̄)2 ≤ c2 t(k, h)2 kqγ − q̄k2L2 (I,Rm ) ≤ c2 t(k, h)2
r2
.
4
Combining the three parts and using the definition of t(k, h), we have
21
T
T
r2
Gkh (qt(k,h) ) ≤ c1 k log + 1 + h2 log + 1 + t(k, h)(c2 t(k, h) − γ)
k
k
4
2
r
= t(k, h)(c4 r2 − γ) + t(k, h)(c2 t(k, h) − γ)
4
= t(k, h) c4 r2 − 2γ + c2 t(k, h)r2 .
22
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
Hence, for h, k are sufficiently small such that 0 < t(k, h) < 1, we obtain from (44)
and the definition of c4 that
Gkh (qt(k,h) ) ≤ t(k, h) c4 r2 − 2γ + c2 r2
and the feasibility of qt(k,h)
≤ (c4 − γ) + (c2 r2 − γ)
γ
3
≤ − γ
2 4
1
≤− γ<0
4
is verified.
The proposition above in particular ensures that Qrkh,feas is not empty once k, h
are small enough.
Corollary 29. For k, h sufficiently small, there exists at least one global solution
r
q̄kh
∈ Qrkh,feas of (Prkh ).
In a second step, we show that the linearized regularity condition of Assumption 10 continues to hold in the discrete setting.
Lemma 30. Under Assumption 10, for k, h small enough it holds
0
1
r
r
r
Gkh (q̄kh
) + Gkh (q̄kh
)(qγ − q̄kh
)≤− γ<0
2
(46)
on I.
0
Proof. In view of Assumption 10, we add and subtract G(q̄), Gkh (q̄), G (q̄)(qγ − q̄)
to obtain
0
0
r
r
r
r
Gkh (q̄kh
) + G (q̄kh
)(qγ − q̄kh
) = G(q̄) + G (q̄)(qγ − q̄) + Gkh (q̄kh
)
0
0
r
r
+ G (q̄kh
)(qγ − q̄kh
) − G(q̄) − G (q̄)(qγ − q̄)
0
r
r
r
≤ −γ + Gkh (q̄kh
) + Gkh (q̄kh
)(q̄ − q̄kh
) − Gkh (q̄)
{z
}
|
(b1 )
0
0
r
+ Gkh (q̄) − G(q̄) + Gkh (q̄kh
) − G (q̄) (qγ − q̄) .
|
{z
} |
{z
}
(b2 )
(b3 )
r
(b1 ) Taylor expansion of Gkh (q̄) at q̄kh
reads
0
1 00
r 2
r
r
r
Gkh (q̄) = Gkh (q̄kh
) + Gkh (q̄kh
)(q̄ − q̄kh
) + Gkh (qζ )(q̄ − q̄kh
) ,
2
r
with qζ convex combination of q̄ and q̄kh
, yielding
1 00
r 2
r
(b1 ) = − Gkh (qζ )(q̄ − q̄kh
) ≤ c2 kq̄ − q̄kh
k2L2 (I,Rm ) ≤ c2 r2 ,
2
where we used Gkh being a C 2 -functional together with the feasibility of
r
q̄kh
for (Prkh ).
(b2 ) By definition of c1 it holds
Z
Gkh (q̄k ) − G(q̄) =
ukh (q̄) − u(q̄) ω(x)dx
Ω
21
T
T
≤ c1 k log + 1 + h2 log + 1 .
k
k
A priori error estimates for state constrained semilinear parabolic optimal control problems
23
(b3 ) By definition of c3 it follows
T
21
r2 T
r
(G0kh (q̄kh
) − G0 (q̄))(qγ − q̄) ≤ c3 k ln + 1 + h2 ln + 1 +
k
k
2
In conclusion, for k, h sufficiently small and thanks to (44), the three estimates for
(b1 ), (b2 ), (b3 ) yield
12
0
T
T
r
r
r
Gkh (q̄kh
) + G (q̄kh
)(qγ − q̄kh
) ≤ −γ + c2 r2 + c1 k log + 1 + h2 log + 1
k
k
21
r2 T
T
+ c3 k log + 1 + h2 log + 1 +
k
k
2
c3 2
≤ −γ + (c2 + )r + (c1 + c3 )·
2
12
T
T
· k log + 1 + h2 log + 1
k
k
12
3
T
T
≤ − γ + (c1 + c3 ) k log + 1 + h2 log + 1
4
k
k
1
≤ − γ.
2
We now introduce the feasible competitors for the continuous auxiliary problem
(Pr ). The proof is similar to Proposition 28, therefore we highlight only the main
arguments.
r
Proposition 31. Let q̄kh
be a global optimum for (Prkh ) and qγ be the Slater’s point
from Assumption 10. Further, let
τ (k, h) =
c1 (k(log(T /k) + 1)1/2 + h2 (log(T /k) + 1)
c4 r2 − γ
be given with a constant c4 such that 0 < c4 r2 − γ < γ/2. Then, the sequence of
controls defined by
(47)
r
r
qτ (k,h) = q̄kh
+ τ (k, h)(qγ − q̄kh
)
is feasible for (Pr ), for k, h sufficiently small.
Proof. Analogously to Proposition 28, with the help of Taylor expansion, this time
r
of Gkh (qτ (k,h) ), at q̄kh
, we obtain
G(qτ (k,h) ) = G(qτ (k,h) ) − Gkh (qτ (k,h) ) +
|
{z
}
(b1 )
0
r
r
r
r
(1 − τ (k, h)Gkh (q̄kh
)) + τ (k, h)(Gkh (q̄kh
) + Gkh (q̄kh
)(qγ − q̄kh
))
|
{z
}
(b2 )
1 00
r
+ Gkh (qζ )(qτ (k,h) − q̄kh
)
2
|
{z
}
(b3 )
≤ τ (k, h) c4 r2 − 2γ + c2 τ (k, h)r2 ,
24
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
r
where we used Theorems 22 and 24 for (b1 ), the feasibility of q̄kh
and discrete Slater
2
condition of Lemma 30 for (b2 ), Gkh being a C -functional together with (43) for
(b3 ).
Then, once k, h are sufficiently small such that 0 < τ (k, h) < 1, in addition to
the prerequisite of Lemma 30, the claim follows as in Proposition 28.
With these results at hands, we now show that global solutions of (Prkh ) converge
to the considered local solution of (P).
Proposition 32. Let k, h be small enough, such that Propositions 28 and 31 hold.
Let q̄ be a local solution for (P) satisfying the assumptions of Theorem 11 and
r
Assumption 13, and let q̄kh
be a global solution of (Prkh ). Then it holds the error
estimate
12
T
T
r
(48)
kq̄ − q̄kh
k2L2 (I,Rm ) ≤ c k log + 1 + h2 log + 1 .
k
k
Proof. Let qt(k,h) and qτ (k,h) be defined as in Proposition 28 and Proposition 31,
respectively, and let k, h be small enough such that 0 < t(k, h), τ (k, h) < 1. We
have
r
r
kq̄ − q̄kh
kL2 (I,Rm ) ≤ kq̄ − qτ (k,h) kL2 (I,Rm ) + kqτ (k,h) − q̄kh
kL2 (I,Rm ) .
For the second term we have
21
12 T
T
r
kqτ (k,h) − q̄kh
kL2 (I,Rm ) ≤ c k log + 1 + h2 log + 1
,
k
k
r
since, thanks to Proposition 31, qτ (k,h) converges strongly in L2 (I, Rm ) to q̄kh
with
order τ (k, h). Therefore, we are left with the first term.
The competitor qτ (k,h) is feasible for (Pr ) and, using the quadratic growth condition (17), we obtain
δkq̄ − qτ (k,h) k2L2 (I,Rm ) ≤ j(qτ (k,h) ) − j(q̄)
r
r
= j(qτ (k,h) ) − jkh (q̄kh
) + jkh (q̄kh
) − jkh (qt(k,h) )
+ jkh (qt(k,h) ) − j(q̄)
r
≤ j(qτ (k,h) ) − jkh (q̄kh
) + jkh (qt(k,h) ) − j(q̄),
|
{z
} |
{z
}
(d1 )
(d2 )
r
where in the last step we have used that qt(k,h) ∈ Qrkh,feas and q̄kh
is a global
r
optimum for (Pkh ).
We now analyze the two terms separately.
(d1 ) With simple algebraic manipulations and the Cauchy-Schwarz inequality,
we have
1
r
r
r
) − 2ud kI ku(qτ (k,h) ) − ukh (q̄kh
)kI
j(qτ (k,h) ) − jkh (q̄kh
) ≤ ku(qτ (k,h) ) + ukh (q̄kh
2
α
r
r
+ kqτ (k,h) + q̄kh
kL2 (I,Rm ) kqτ (k,h) − q̄kh
kL2 (I,Rm ) .
2
Then, by means of the stability of the solution u and ukh of (2) and (24),
respectively, together with the boundedness of Qad , and with the help of
A priori error estimates for state constrained semilinear parabolic optimal control problems
25
the Cauchy-Schwarz inequality, we get
r
r
r
r
j(qτ (k,h) ) − jkh (q̄kh
) ≤ c ku(qτ (k,h) ) − u(q̄kh
)kI + ku(q̄kh
) − ukh (q̄kh
)kI
r
+ kqτ (k,h) − q̄kh
kL2 (I,Rm )
r
r
r
≤ c ku(q̄kh
) − ukh (q̄kh
)kI + kqτ (k,h) − q̄kh
kL2 (I,Rm ) ,
where in the last step we have used (7a).
The first term is a discretization error that can be estimated by [31,
Theorems 3.3 and 4.2] together with the regularity of the solution of (2),
obtaining
r
r
ku(q̄kh
) − ukh (q̄kh
)kI ≤ c(k + h2 ).
r
The estimate for the second term, kqτ (k,h) − q̄kh
k, follows directly from
Proposition 31. Summing up, we conclude
12
T
T
j(qτ (k,h) ) − jk (qkr ) ≤ c k + h2 + k log + 1 + h2 log + 1
k
k
12
T
T
2
≤ c k log + 1 + h log + 1 .
k
k
(d2 ) We proceed exactly as for (d1 ).
1
kukh (qt(k,h) ) + u(q̄) − 2ud kI kukh (qt(k,h) ) − u(q̄)kI
2
α
+ kqt(k,h) + q̄kL2 (I,Rm ) kqt(k,h) − q̄kL2 (I,Rm )
2
jkh (qt(k,h) ) − j(q̄) ≤
≤ c kukh (qt(k,h) ) − u(qt(k,h) )kI + kqt(k,h) − q̄kL2 (I,Rm )
21
T
T
≤ c k log + 1 + h2 log + 1 .
k
k
Combining (d1 ) with (d2 ), we have the assertion.
It is readily seen that, for k, h small enough, global solutions of (Prkh ) are local
r
solutions of (Pkh ), as the constraint kq̄−q̄kh
kL2 (I,Rm ) ≤ r is not active. In particular,
this ensures the existence of a sequence q̄kh , of local solutions to (Pkh ), converging
to q̄. We formalize this in the main result of the paper
Theorem 33. Let q̄ be a local solution of (P) satisfying the assumptions of Theorem 11 and Assumption 13. Then, for k, h sufficiently small, there exists a sequence
(q̄kh ) of local solution of (Pkh ) converging to q̄ as k, h → 0. Further, there holds the
error estimate
!
12
T
T
2
2
(49)
kq̄ − q̄kh kL2 (I,Rm ) ≤ c k log + 1 + h log + 1 .
k
k
Remark 34. Making use of the same technique one can derive an error estimate
arising from the discretization in time only. As one might expect, this reads
12
T
kq̄ − q̄k k2L2 (I,Rm ) ≤ ck log + 1
k
where q̄k is a suitable sequence of local solutions to (Pk ). On the other hand, the
error between the semidiscrete and discrete solution will only satisfy the estimate
26
FRANCESCO LUDOVICI, IRA NEITZEL, AND WINNIFRIED WOLLNER
in Theorem 33. In order to obtain
T
kq̄k − q̄kh k2L2 (I,Rm ) ≤ ch2 log + 1 ,
k
r
a quadratic growth condition in the solution of Pk needs to hold uniformly in k. To
this end, the SSCs need to be transfered to the semidiscrete setting. The difficulty
in this procedure lies in the convergence of the critical directions. As it is not clear
how this can be shown, one would resolve to utilize a strong SSC; thereby avoiding
the need for certain critical directions.
Acknowledgments
The authors are grateful for the support of their former host institutions. To this
end, I. Neitzel acknowledges the support of the Technische Universität München
and F. Ludovici and W. Wollner the support of the Universität Hamburg.
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(F. Ludovici) Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr.
15, 64293 Darmstadt, Germany
E-mail address: [email protected]
(I. Neitzel) Institut für Numerische Simulation, Rheinische Friedrich-Wilhelms-Universität
Bonn, Wegelerstr. 6, 53115 Bonn, Germany
E-mail address: [email protected]
(W. Wollner) Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr.
15, 64293 Darmstadt, Germany
E-mail address: [email protected]